Mathematics and AfL

• Black, P., Harrison, C., Lee, C., Marshall, B. and Wiliam, D. (2002) Working Inside the Black Box: Assessment for Learning in the Classroom. London: King’s College.
• Leahy, S., Lion, C., Thompson, M. and Wiliam, D., 2005 Classroom Assessment: Minute by Minute, Day by Day, Educational Leadership, Vol 63(3)
• Wiliam, D., (2011) Embedded Formative Assessment. Bloomington IN, Solution Tree Press

When you have found out that a child cannot say "count backwards from twenty" you must have several ideas ready to help them to grow their understanding.

Have you used

• A collection of like objects?
• A collection of unlike objects?
• A number line?
• Cuisenaire rods?
• A game? such as?
• Etc.

"...teachers do not create learning; only learners create learning. And yet our classrooms seem to be based on the opposite principle - that if they try really hard, teachers can do the learning for the learners." (Wiliam 2011 p.145)

For learners to change 'do you mean improve/?their trajectory in learning, they must receive task-related feedback from teachers and peers. Feedback within formative assessment not only identifies areas for improvement, but must be provided with a view to enabling learners to make necessary improvements in their work; pertinent feedback is an essential element in promoting learning.

Assessment must be part of learning because teaching is not the same as learning:

Consider the following extract adapted from Denvir and Brown (1986):

One student was assessed as having the following 'gaps' in her knowledge.

She does not know:

Being able to talk about these operations seems to be important in thinking about and doing these operations (see for example Anna Sfard (2001)). If the child is unable to talk about and model these ideas (e.g. on a number line) then their understanding may be operational and not relational (Skemp 1976), they may be parroting not thinking. There is a great deal more vocabulary that must be secure before moving onto multiplication and division.

Firstly children must be secure when counting and must understand 'more than' and 'less than'. They must be able to count on and count back and have developed the vocabulary to do so.

Assessment for learning and formative assessment tend to be used interchangeably. There is some disagreement about how formative and summative assessment are defined and that is in part due to the breadth of actions that can be considered formative. Summative assessments can, without doubt, be used formatively ( Black et al 2002). There is some agreement that the process of using assessments formatively: